Constructs of Symmetry in Islamic Art

by Terry Allen

Full Table of Contents

Section One. Types of Symmetry

Symmetry is a characteristic of nature; it is not something that has been invented by artists and architects. Symmetry has been well investigated by mathematicians and crystallographers, for example, and there is nothing for art historians to add to their work; it can be used straightforwardly.

Symmetry comes in various forms no matter how it is analyzed.

Traditional Mathematical Approaches to Symmetry

A brief survey of some of the literature that is understandable by nonmathematicians, or at least was understandable by me, may serve to introduce the mathematical view of symmetry. I found that it was useful in understanding any particular point to read several explanations of it.

Symmetry, by Hermann Weyl, is much cited as a classic.1 Other mathematicians have followed him in tracing uses of symmetry in art and in tracing occurrences of symmetry in nature and across mathematical domains. His approach to actually explaining symmetry is progressive rather than expository.

Transformation Geometry: An Introduction to Symmetry, by George E. Martin, an undergraduate textbook, provides a rigorous treatment of the relevant topics;2 Connections: The Geometric Bridge between Art and Science, by Jay Kappraff, is similarly rigorous but not quite as formal.3

Tilings and Patterns, by Branko Grünbaum and G.C. Shephard, is a broad and deep attempt to establish a comprehensive mathematical description of tilings and patterns (apparently not much dealt with previously from a mathematical perspective), with ample bibliography.4 It is avowedly written for students of mathematics and professional mathematicians, conceding little to “non-mathematicians whose interests include patterns and shapes (such as artists, architects, crystallographers and others),” the authors's protestations to the contrary.5 Here I am concerned directly with neither tilings nor patterns, but with symmetry; I am sure there is much in Grünbaum and Shephard's book to be mined by the humanist, but it will have to be mined by others. For those interested in trying, I suggest focusing on those sections relating to tilings and patterns that actually appear in art and architecture, disregarding the general discussion of possible patterns that no one has ever invented and impossible (except for the geometrician) tilings composed of tiles that extend infinitely in various directions. As for symmetry, Grünbaum and Shephard discuss isometries in general and with regard to tilings usefully.6

Symmetries of Culture: Theory and Practice of Plane Pattern Analysis, by Dorothy K. Washburn and Donald W. Crowe, is a thoroughgoing handbook explaining how to analyze two-dimensional patterns on archaeological and anthropological artifacts in strictly mathematical terms.7 Washburn and Crowe accessibly provide historical and mathematical background as well as a methodology for analysis (using flow charts, which is to say, proceeding step by step through sets of questions that can be answered yes or no).8 They point out that symmetry analysis (their term) of repeating patterns was advocated by anthropological archaeologists as early as the nineteen-forties.

Wasmaʿa Chorbachi made a specific plea for the use of mathematical terminology for symmetrical patterns in Islamic art in 1989 in “In the Tower of Babel: Beyond Symmetry in Islamic Design”,9 which also contains the most forthright remarks I have seen in print regarding the corrosive effect on the study of Islamic art of large amounts of money flowing from the Muslim world to Western scholars and institutions.

George Hersey, in Architecture and Geometry In the Age of the Baroque, remarks “unfortunately, architectural historians and critics almost never write about symmetry,” and goes on to demonstrate simple forms of symmetry in sometimes idiosyncratic interpretations.10 Certain basic concepts are presented simply in an online article sponsored by the Math Forum and the Textile Museum, “Symmetry and Pattern: The Art of Oriental Carpets”.

Symmetry can be a property of plane figures (such as regular polygons); one-dimensional patterns, which are symmetrical in one direction only (such as an undulating border pattern); and two-dimensional patterns, which are symmetrical in more than one direction (such as a checkerboard). These two-dimensional patterns can be tiling patterns, in which one shape completely covers the plane, or not. There is an additional complication to two-dimensional patterns, known as color symmetry, which I attempt to explain below. I ignore three-dimensional symmetry here.

I am certain I have made errors in the following two sections, but nothing ventured, nothing gained—I will be thankful to readers who point them out to me.

Isometries

A plane figure or set of figures can have multiple symmetries. A symmetry in such a figure or figures is the property of remaining invariant through some transformation (that is, and phrased informally, the distance between any two points in the figure or set of figures remains the same after the transformation).11 Such a transformation “leaves the figure looking exactly the same … as it did before”.12 There are four basic such transformations, which are known as isometries (also discussed as “motions”).13 Alternately, symmetry is attributed to the motions rather than the figures moved, from which point of view a symmetry of a figure or set of figures is a motion that leaves the figure invariant. A special case is the identity transformation: if a figure is moved so that every point ends up in the same place as before (for example, if a circle is rotated 360° through its center), it is the same as it was before. 14

By contrast, stretching a figure along some axis, or enlarging it, is a transformation but not an isometry.15

The first transformation or motion listed here, reflection, can, if repeated no more than three times, produce the effect of any of the other three, but the results of a single application of each of the four types differ.16

reflection

A reflection takes place along an axis running through the figure (or set of figures, or pattern). A figure has reflection symmetry if it is invariant after reflection (more colloquially, if it appears the same after reflection). Neither nor has reflection symmetry, but the pair 

    

does. This pair has a single axis along which it has reflection symmetry; an equilateral triangle has three; a square has four; a circle has infinitely many;17 two circles next to each other have one. Each of these symmetrical reflections counts as a symmetry, so the figures listed have respectively one, three, four, an infinite number, and one reflection symmetries.

translation

Translation is the operation of moving a figure in a single direction. No single (or “finite”) figure has translational symmetry, but an infinitely long and evenly spaced row of identical figures does: it can be moved by one or more multiples of the distance between like points in the identical figures and remain the same. As there are no such rows of figures in real life, the mathematical concept of translation must be adapted to finite rows of figures, such as colonnades or border patterns. Here is an attempt to illustrate translation symmetry: 

 …        …

rotation

Rotation in a plane is around a fixed point, the center of rotation, whether that point is part of the figure rotated or outside it. Any regular polygon (equilateral triangle, square, regular pentagon) has rotational symmetry (around its center point). So does a regular swastika. A regular polygon has the same number of rotational symmetries as it has reflection symmetries.18

Note that a figure can be rotated by any amount. Also, if the center of rotation lies outside the figure, the result of the rotation can result in a second figure lying at any angle to the original, for example (and rather roughly): 

 
   

See also the discussion below of rotocenters.

glide reflection

Glide reflection is “a translation (‘glide’) followed by a reflection in a line parallel to the direction of translation”19 or the same two transformations in reverse order, like a line of footprints, the right footprints offset from the left along the line of travel. No single (“finite”) figure has glide reflection symmetry (because it does not have translational symmetry). Again, to adapt this notion to real life one must be willing to consider rows of offset figures (footprints, border patterns again) as qualifying: 

  …          …
…          …

These four types of isometry are simple enough to understand and describe that there is really no reason to use any other terminology for them, even in art historical studies. (Note, however, that they do not suffice for describing tiling patterns, as the the section “Field Patterns” in the Math Forum article shows.)

Group Theory and Tilings

Group theory involves the concept of symmetry groups, which are not difficult to understand given a grasp of isometries. “Given any figure, the symmetry group of that figure is the collection of all transformations that leave that figure invariant”;20 that is, the collection of all its symmetries.

Group theory, arising from the consideration of symmetry groups, has achieved a quite abstract level and is applied to all kinds of groups, not just symmetry groups; I do not pretend that group theory is relevant to art history. For the study of tiling or “wallpaper” patterns (in which a single repeated shape covers the entire plane), however, it is of some interest that any pattern that has translational symmetry has one of only seventeen distinct symmetry groups (if one ignores complications such as alternating colors). Such patterns, which can be extended indefinitely in the imagination (that is, they have unendliche Rapport21) are common in Islamic art and architecture. (Also common are patterns that cover the plane completely using more than one repeated shape, like the first example from the Alhambra, below.)

Washburn and Crowe call these “one-color, two-dimensional patterns”. If the pattern has two colors that are distributed regularly among its elements and a transformation either moves all elements of one color onto elements of the other (as with a chessboard rotated 90°) or moves all elements onto like-color elements, it is a “two-color” pattern.22

As a side note, the translational symmetry of tiling patterns can also be analyzed in terms of “Dirichlet domains”. For a point in a lattice of points that correspond along two axes, which need not be at right angles as in the example below (nor need the points be the same distance from each other in both axes) 

          .  .  .  .  .  .  .  .
          .  .  .  .  .  .  .  .
          .  .  .  .  .  .  .  .
          .  .  .  .  .  .  .  .
          .  .  .  .  .  .  .  .
          .  .  .  .  .  .  .  .
          .  .  .  .  .  .  .  .
          .  .  .  .  .  .  .  .

a Dirichlet domain is “the entire region of the plane that is nearer to that point than to any other lattice point” (Devlin, op. cit., p. 163). These regions (domains) likely cut across the actual units of design. There are only five distinct shapes of Dirichlet domain, all quadrilaterals or hexagons. Generalizing this notion of lattices to three dimensions leads fairly directly to crystallography, but is not, I believe, relevant to the history of Islamic art, and for that matter, what interests most art historians about tiling patterns is not the shape of their Dirichlet domains but the shapes of the units composing the pattern.

Notations for Patterns

There are multiple crystallographic notations for patterns (plane symmetry groups), which I shall not recapitulate here.23 Washburn and Crowe prefer a particular commonly used set of notations that so far as I can see seems well chosen, but for two-dimensional patterns these notations are unfortuately cryptic (even if one studies how they are constructed: each element has a meaning) and vulnerable to typographical error, such as p2mg or pb'gg. But they are certainly preferable to homegrown terminology such as the Math Forum article uses (e.g., “rotations (2) + reflections + reflections”). On the one hand it may be more important to know whether two patterns share the same classification than to be able to give it a memorable name; on the other, the characteristics of a pattern that are captured in the structure of a notation are disclosed by working with Washburn and Crowe's flow charts, which are more interesting than they may seem at first.


1. Ibid.

2. New York, 1982.

3. New York, 1991.

4. New York, 1987; to be distinguished from an abridged version published later under much the same title. I have not seen the abridgement, and it may contain the material I discuss here.

5. Ibid., p. vii. For example, “For any isometry σ and any set S we write σS for the image of S under σ.”

6. Ibid., pp. 26–45.

7. Seattle, 1988. See also Clare E. Horne, Geometric symmetry in patterns and tilings, Cambridge, 2000.

8. This is not to say that using the flow charts is particularly easy; one must work back and forth through the text (which is not rigidly consistent in its terminology) to understand just what the questions mean. Studying their examples can help, although studying the rules governing their notation may not.

9. Unfortunately buried from view by art historians in Symmetry 2: Unifying Human Understanding, ed. István Hargittai, Oxford, 1989, pp. 751–89.

10. Chicago, 2000, p. 99.

11. For a formal definition see Martin, op. cit., p. 26: for the geometrician, a transformation of a figure produces a new set of points; the distance that remains the same is then between points in the original figure and the corresponding points in the figure after transformation.

12. Keith Devlin, Mathematics: The Science of Patterns, New York, 1994, repr. 2003, p. 146.

13. See Washburn and Crowe, op. cit., pp. 44ff., which I find slightly difficult; their way of describing these transformations, also used by Weyl, is that a plane figure is symmetric with respect to a given line if it is “carried into itself” by reflection in that line (as opposed to changing it into a different figure; Weyl, op. cit., p. 43). They also write of “motions of the plane”. Martin, op. cit., p. 18, points out that no motion is actually involved; “transformations are just one-to-one correspondences among the points” of figures. One need not worry about how to manipulate a plane of infinite extent.

14. Kappraff, op. cit., p. 390, and Grünbaum and Shephard, op. cit., p. 26, seem to treat it as a distinct isometry, but it is often ignored as a type as uninteresting except for set theory.

15. Martin, op. cit., p. 1.

16. Martin, op. cit., ch. 5.

17. See Weyl, op. cit., p. 46 for the circle.

18. Counting a 360° rotation around its center, which is the identity isometry; Grünbaum and Shephard, op. cit., p. 27, fig. 1.3.2.

19. Washburn and Crowe, op. cit., p. 50.

20. Devlin, op. cit., p. 146.

21. See Terry Allen, Five Essays on Islamic Art, Sebastopol, Calif., 1988, chapter 1.

22. Op. cit., p. 57, 63ff. Washburn and Crowe discuss briefly but do not consider in detail patterns involving three and more colors.

23. See Chorbachi, op. cit., 774–75, who does not express a preference except for uniformity in use of one notation or another.